Question

Find the average rate of change of each function on the interval specified in simplest form. Do not type your answer in factored form and do not type any spaces between characters. \frac{f(x+h)-f(x)}{h} given f(x)=2x^2-3x on the interval [x,x+h]The average rate of change is Answer

Answer 1

Given:

`f(x)=2x^2-3x`

To find the average rate of change on the interval [x, x+h], we use the operation as defined below;

`(f(x+h)-f(x))/(h)`

Hence,

`\begin{gathered} \text{Solving for f(x+h),} \\ f(x)=2x^2-3x \\ f(x+h)=2(x+h)^2-3(x+h) \\ f(x+h)=2(x+h)(x+h)-3x-3h \\ f(x+h)=2(x^2+hx+hx+h^2)-3x-3h \\ f(x+h)=2(x^2+2hx+h^2)-3x-3h \\ f(x+h)=2x^2+4hx+2h^2-3x-3h \end{gathered}`

Substituting it into the operation,

`\begin{gathered} (f(x+h)-f(x))/(h) \\ =(2x^2+4hx+2h^2-3x-3h-(2x^2-3x))/(h) \\ S\text{ implifying the solution further,} \\ =(2h^2+4hx-3h+2x^2-3x-2x^2+3x)/(h) \\ =(2h^2+4hx-3h+2x^2-2x^2+3x-3x)/(h) \\ =(2h^2+4hx-3h)/(h) \\ \text{Factorizing the common factor out from the numerator,} \\ =(h(2h+4x-3))/(h) \\ =2h+4x-3 \\ (f(x+h)-f(x))/(h)=4x-3+2h \end{gathered}`

Therefore, the average rate of change is;

`4x-3+2h`